Multivariate stochastic equation: \[\begin{gather*}
\color{red}{\mathrm{d}} \boldsymbol{\eta}_i(\color{red}{t})=\left(\mathbf{A} \boldsymbol{\eta}_i(\color{red}{t})+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \color{red}{\mathrm{d} t}+\mathbf{G} \color{red}{\mathrm{d}} \mathbf{W}_i(\color{red}{t})
\end{gather*}\] modelling change in latent variables \(\boldsymbol{\eta}_i(t)\) as a function of time over infinitesimal small time steps \(\color{red}{\mathrm{d} t}\)
Recap II
Multivariate stochastic differential equation: \[\begin{gather*}
\mathrm{d} \boldsymbol{\eta}_i(t)=\left(\color{red}{\mathbf{A}} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t)
\end{gather*}\] modelling change as a function of time over infinitesimal small time steps \(\mathrm{d} t\)
\(\color{red}{\mathbf{A}}\): drift matrix with auto effects on the diagonal and cross effects on the off-diagonals (\(\mathbf{A} \in \mathbb{R}^{v \times v}\)).
Recap II
Multivariate stochastic differential equation: \[\begin{gather*}
\mathrm{d} \boldsymbol{\eta}_i(t)=\left(\mathbf{A} \boldsymbol{\eta}_i(t)+\mathbf{B} \mathbf{z}_i+\mathbf{M} \boldsymbol{\chi}_i(t)\right) \mathrm{d} t+\mathbf{G} \mathrm{d} \mathbf{W}_i(t)
\end{gather*}\] modelling change as a function of time over infinitesimal small time steps \(\mathrm{d} t\)
\(\mathbf{A}\): drift matrix with auto effects on the diagonal and cross effects on the off-diagonals (\(\mathbf{A} \in \mathbb{R}^{v \times v}\)).
\(\mathbf{z_i}\): time independent predictors
\(\boldsymbol{\chi}_i(t)\): time dependent predictors
\(\mathbf{W}_i(t)\): stochastic error term (Wiener process)